Combinatorics

Use the properties of binomial coefficients and the Binomial Theorem to solve:

1. a) Fully expand the expression $(x + 2)^4$.

1. b) Without expanding the whole binomial, calculate the value of the coefficient of the term containing $x^3$ in the expansion of $(x + 3)^5$.

The figure shows the rows of Pascal's triangle up to $n = 5$. Its numbers are the binomial coefficients $\binom{n}{k}$: remember that the coefficient of a term of the binomial, besides that number, also includes the powers of the second term.

In the expansion of a new binomial, an engineering student needs to determine certain coefficients. Consider the binomial expansion of $Q(x) = \left( \dfrac{x^2}{2} - \dfrac{k}{x} \right)^{12}$, where $k$ is a positive real number.

1. a) Determine the value of $k$ knowing that the term independent of $x$ in the expansion of $Q(x)$ is equal to $7920$.
2. b) Having found $k$, calculate the coefficient of the term containing $x^9$.

A 6th Year Leaving Certificate student wants to set up a $4$-digit access code with different digits for their gym locker. They decide to use only the odd digits $\{1, 3, 5, 7, 9\}$ and cannot repeat any digit.

1. a) How many different codes can they create?

1. b) If the student forgets the code but remembers that the first digit was a $7$, how many different codes would they have to try at most?

A laboratory has 8 precision temperature sensors and 6 humidity sensors. For an environmental control experiment, 4 sensors must be selected at random.

1. a) In how many different ways can the selection of 4 sensors be made if order does not matter?
2. b) What is the probability that exactly 2 temperature sensors and 2 humidity sensors are selected?
3. c) What is the probability that all selected sensors are of the same type?

A municipal electric bicycle rental service in an Irish city is renewing its smart locks. Each user must set a $4$-digit PIN code with two conditions: the first digit cannot be $0$, and no two consecutive digits can be the same (for example, the code $4727$ is valid, but $4772$ is not).\n\na) Before performing any mathematical calculation, intuitively estimate whether the number of valid codes will be closer to $5\,000$, to $6\,500$, or to $9\,000$. Briefly justify the basis of your estimate.\nb) Apply the fundamental counting principle to calculate exactly how many different codes satisfy both conditions.\nc) Check whether the exact result obtained supports your initial estimate, and calculate what percentage it represents relative to the total number of four-digit combinations that do not start with zero (from $1\,000$ to $9\,999$).

To elect the student delegation that will represent a high school on the local Student Council, $3$ students must be selected from a group of $24$ volunteers from 6th Year. Cian reasons as follows on his exam: «For the first spot I have 24 options, for the second 23, and for the third 22, so the total number of possible delegations is $V_{24,3} = 24 \times 23 \times 22 = 12\,144$».\n\na) Identify the conceptual error made by Cian based on the two key questions of combinatorics: does order matter in this case? Can elements be repeated?\nb) Correct the calculation and find the real number of possible delegations, justifying which operation Cian's formula must be replaced with.\nc) If the three spots had differentiated roles (a spokesperson, a secretary, and a committee member), would Cian's original calculation be correct? Justify your answer.

In a robotics championship, a team must select $3$ representatives from a group of $8$ students to present the project to the panel of judges.

1. a) In how many different ways can the group of representatives be chosen if the selection order does not matter?

1. b) If among the $8$ students there are $2$ who specialise in programming and it is decided that at least one of them must be in the group, how many different selections are possible?

A streaming platform is designing a new custom interface. For the user profile, an avatar can be chosen by combining the following elements: 4 different face shapes, 5 types of hairstyle and 3 styles of glasses. In addition, the username must be a 5-character string consisting of 3 lower-case letters (from a 26-letter alphabet) followed by 2 digits.

1. a) How many different avatars can be created?
2. b) How many distinct usernames can be generated if neither letters nor digits can be repeated?
3. c) If a user decides to choose their avatar and username at random (under the conditions of part b), what is the probability that their name starts with the letter "z" and ends with the digit "9"?

To grade the Leaving Certificate Higher Level Mathematics exams at a State Examinations Commission centre, an examining board of $6$ examiners must be formed. The available faculty consists of $10$ teachers: $6$ come from non-fee-paying secondary schools and $4$ from fee-paying private secondary schools. The organising committee's regulations require that the board have at least $2$ teachers from non-fee-paying schools and at least $2$ from fee-paying private schools.\n\na) Model the situation by identifying all valid cases for the composition of the board according to the number of teachers from each type of school.\nb) Calculate how many different boards can be formed in compliance with the established regulations.\nc) If the $6$-person board were chosen completely at random from the $10$ teachers (without requiring any representation quota), what would be the probability that the board complies with the legal regulations? Interpret whether the restriction excludes many or few of the possible combinations.

Let $A$ and $B$ be two events associated with a random experiment such that $P(A) = 0.6$, $P(B) = 0.4$ and $P(A \cap B) = 0.2$. Calculate the following probabilities:

1. a) $P(A \cup B)$

1. b) $P(\bar{A} \cap \bar{B})$ (probability that neither $A$ nor $B$ occurs).

1. c) $P(A - B)$ (probability that $A$ occurs but $B$ does not occur).

The Venn diagram shows the events $A$ and $B$ within the sample space $E$, with the probability of the intersection.

In a robotics competition, teams must pass two independent tests. It is known that the probability of a robot passing the speed test is $P(V) = 0.75$ and the probability of passing the accuracy test is $P(P) = 0.60$. Calculate the probability that a randomly chosen team:

1. a) Passes at least one of the two tests.
2. b) Passes the speed test but not the accuracy test.
3. c) Given that it has passed at least one test, what is the probability that it has passed both?

A cybersecurity team is analysing the vulnerability of an access system that uses passwords of $8$ characters. The allowed characters are the $26$ letters of the alphabet (case-insensitive) and the digits from $0$ to $9$. A password is considered "strong" if it meets the following conditions:

1. It contains exactly $2$ distinct numbers and $6$ letters.

2. The letters cannot be repeated.

3. The two numbers must be in consecutive positions.

Calculate:

1. a) The total number of strong passwords that can be generated.
2. b) If brute-force software generates a random password following only the structure of $8$ characters (letters or numbers), what is the probability that it guesses a specific strong password on the first attempt?

A vehicle registration system in Ireland assigns each vehicle a license plate consisting of a block of $4$ digits (from $0000$ to $9999$) followed by a block of $3$ consonant letters chosen from $20$ possible letters (the vowels A, E, I, O, U, and the letters Ñ and Q are excluded to avoid confusion and inappropriate combinations).\n\na) Calculate the total number of theoretical license plates that this system can issue and express the result using powers of base ten.\nb) A combinatorics enthusiast looks for "symmetric license plates": those whose $4$-digit numerical block is a palindrome (for example, $4884$ or $0330$) and whose $3$-letter block is a palindrome (the first and third letters are the same, as in $TRT$). Determine how many symmetric license plates exist and find the probability that a randomly registered vehicle has this property.\nc) The Road Safety Authority (RSA) is investigating a license plate reading algorithm and detects errors in plates that it calls "heterogeneous mixed": those that have their $4$ digits strictly distinct from each other and, simultaneously, contain exactly two identical letters and one different letter in the letter block (in any order, as in $LBL$ or $DDH$). Model and calculate how many "heterogeneous mixed" license plates exist in the potential Irish vehicle fleet.

A laboratory has 8 precision temperature sensors and 6 humidity sensors. For an environmental control experiment, 4 sensors must be selected at random.

1. a) In how many different ways can the selection of 4 sensors be made if order does not matter?
2. b) What is the probability that exactly 2 temperature sensors and 2 humidity sensors are selected?
3. c) What is the probability that all selected sensors are of the same type?

A competitive examination consists of $80$ topics. The candidate studies $55$ topics chosen at random. The examination board draws $3$ topics by lottery and the candidate must choose one to write on.

1. a) What is the probability that the candidate does not know any of the three drawn topics?
2. b) What is the probability that they know exactly two of the topics?
3. c) If the candidate passed the examination (which implies they knew at least one of the topics), what is the probability that they knew all three drawn topics?

In a study on the use of mobile devices in a class of $30$ students, the following contingency table was obtained, classifying students by their preferred operating system and whether they regularly shop online:

If a student is chosen at random, calculate:

1. a) The probability that they use iOS.

1. b) The probability that they make online purchases given that they use Android.

1. c) Are the events "Using Android" and "Making online purchases" independent? Justify your answer.

A survey of 250 students at a technology centre was carried out regarding their use of operating systems. The results are shown in the following contingency table:

If a student is chosen at random, calculate:

1. a) The probability that they are a Higher Vocational student and mainly use Linux.
2. b) Given that the student mainly uses Windows, what is the probability that they are an Intermediate Vocational student?
3. c) Are the events "being a Higher Vocational student" and "mainly using Linux" independent? Give reasons for your answer.

A video game server has two connection nodes: Node 1, which serves $70\%$ of the users, and Node 2, which serves the rest. It is known that $5\%$ of the connections on Node 1 suffer lag, while on Node 2 this problem occurs in $10\%$ of the connections.

1. a) Represent the situation using a probability tree diagram.

1. b) If a player joins the server at random, what is the probability that their connection suffers lag?

Three phone operators, $M_1$, $M_2$ and $M_3$, handle $40\%$, $35\%$ and $25\%$ of customer service call centre calls, respectively. The percentage of calls that drop unexpectedly for each operator is $2\%$, $3\%$ and $4\%$, respectively.

A call is chosen at random and turns out to have dropped. Calculate the probability that it was handled by operator $M_1$.

A streaming platform uses a recommendation algorithm to predict whether a user will like a new series. The algorithm sometimes fails: $10\%$ of the series that a user would love are not recommended ($FN$: false negative), and $20\%$ of the series that a user would dislike are mistakenly recommended ($FP$: false positive).

If $60\%$ of all series in the catalog are potentially to the taste of a specific user, calculate:

1. a) The probability that the algorithm recommends a randomly chosen series.
2. b) If the algorithm has just recommended a series to this user, what is the probability that it is indeed a series they like?

An electronic components factory has three production lines: $L_1$, $L_2$ and $L_3$. Line $L_1$ produces twice as much as $L_2$, and $L_3$ produces the same amount as $L_1$ and $L_2$ combined. The percentage of defective components is $1\%$ in $L_1$, $3\%$ in $L_2$ and $2\%$ in $L_3$.

1. a) Calculate the probability that a component chosen at random from the total production is defective.
2. b) If we select a component and it turns out to be defective, what is the probability that it comes from line $L_3$?
3. c) If $10$ components from line $L_1$ are analysed independently, what is the probability that at least one of them is defective?

In an electronics factory, the probability that a microchip is defective is $0.02$. If the microchips are packed in boxes of $10$ units and the presence of defects in one is independent of the others:

1. a) What is the probability that there are no defective microchips in a box?

1. b) What is the probability that there is exactly one defective microchip in the box? (Express the result using binomial coefficients).

A 3D printing electronics manufacturing company has observed that the probability of a part having a finishing defect is $5\%$. If a batch of 12 parts is printed:

1. a) What is the probability that exactly 2 parts are defective?
2. b) What is the probability that there is more than one defective part?
3. c) Calculate the mean and standard deviation of the number of defective parts in the batch.

In a genetics laboratory, the resistance of a plant to a pest is studied. It is known that the probability of a plant being resistant is $p$. $8$ plants are chosen at random and it is observed that the probability that exactly $2$ are resistant is equal to the probability that exactly $3$ are resistant.

1. a) Calculate the value of the probability $p$.
2. b) With the value of $p$ found, calculate the probability that at least one plant is resistant.
3. c) Determine the expectation and standard deviation of the number of resistant plants in a sample of $100$ plants.

The range of a new fleet of electric scooters for urban use follows a normal distribution with mean $35$ km and standard deviation $4$ km.

1. a) Calculate the probability that an e-scooter chosen at random has a range greater than $40$ km.
2. b) What percentage of e-scooters will have a range between $30$ and $38$ km?
3. c) If the company wants to ensure that $90\%$ of its e-scooters exceed a certain minimum range $d$, what must be the value of $d$?

The weight of coffee packets produced by a factory follows a normal distribution $N(\mu, \sigma)$. It has been observed that $15.87\%$ of the packets weigh more than $510$ grams, and that $2.28\%$ of the packets weigh less than $480$ grams.

1. a) Calculate the mean $\mu$ and standard deviation $\sigma$ of the distribution. (Recall that for a $Z \sim N(0,1)$, $P(Z < 1) \approx 0.8413$ and $P(Z < 2) \approx 0.9772$).
2. b) If the net weight indicated on the label is $500$ grams, what percentage of the packets is below the legal weight?